Well, these were old-timey bad guys. Maybe they all said something like "curses! foiled again!" when identified? :)
<rimshot />
In all seriousness, though: Once they'd caught their suspect and thrown them in Newgate Prison, the authorities could always summon people from the suspect's hometown to ID them more precisely. In theory. In practice, that would surely have been very expensive and slow, and I have no idea how diligent they really were about all that pesky "legal procedure" stuff, particularly if the prisoner didn't have the money to hire a lawyer. Back then they charged prisoners for everything: you had to pay room and board in prison, so if you hadn't managed to hang onto your money (which was probably pretty difficult when you were being arrested by freelance thief-takers...) and you didn't have local friends to look you up in prison and help pay your bills, you could be stuck in there for life, like Charley on the MTA.